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While there seems to be no proof that the ((6,2,3)) 2 degenerate code has a larger K than any nondegenerate code with n=6 and δ=3, some support comes from the fact that we performed an exhaustive search of all graphs with 6 vertices and did not find a nondegenerate graph code with δ=3 and K>1. But the notion that this degenerate code is superior to nondegenerate codes is undercut by the observation that the well known nondegenerate ((5,2,3)) 2 code uses only five instead of six qubits to achieve equal values of K and δ.
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While there seems to be no proof that the ((6,2,3)) 2 degenerate code has a larger K than any nondegenerate code with n=6 and δ=3, some support comes from the fact that we performed an exhaustive search of all graphs with 6 vertices and did not find a nondegenerate graph code with δ=3 and K>1. But the notion that this degenerate code is superior to nondegenerate codes is undercut by the observation that the well known nondegenerate ((5,2,3)) 2 code uses only five instead of six qubits to achieve equal values of K and δ.
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We omit the details. In some but not all cases one can use the partition theorem with V1 and V2 the center and the peripheral vertices. Allowing some double edges when D>2 extends the range of n values where the partition theorem can be employed.
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We omit the details. In some but not all cases one can use the partition theorem with V1 and V2 the center and the peripheral vertices. Allowing some double edges when D>2 extends the range of n values where the partition theorem can be employed.
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Since the distance δ=3 does not exceed the diagonal distance Δ′ =3 for this graph, a graph code is necessarily nondegenerate, see Sec. 3, and hence the quantum Hamming bound-see p. 444 of Ref. -extended to D=3 applies, and this yields an upper bound of K≤1990.
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Since the distance δ=3 does not exceed the diagonal distance Δ′ =3 for this graph, a graph code is necessarily nondegenerate, see Sec. 3, and hence the quantum Hamming bound-see p. 444 of Ref. -extended to D=3 applies, and this yields an upper bound of K≤1990.
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